Positivity

, Volume 22, Issue 1, pp 245–260 | Cite as

Finite elements in some vector lattices of nonlinear operators

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Abstract

We study the collection of finite elements \(\Phi _{1}\big ({\mathcal {U}}(E,F)\big )\) in the vector lattice \({\mathcal {U}}(E,F)\) of orthogonally additive, order bounded (called abstract Uryson) operators between two vector lattices E and F, where F is Dedekind complete. In particular, for an atomic vector lattice E it is proved that for a finite element in \(\varphi \in {\mathcal {U}}(E,{\mathbb {R}})\) there is only a finite set of mutually disjoint atoms, where \(\varphi \) does not vanish and, for an atomless vector lattice the zero-vector is the only finite element in the band of \(\sigma \)-laterally continuous abstract Uryson functionals. We also describe the ideal \(\Phi _{1}\big ({\mathcal {U}}({\mathbb {R}}^n,{\mathbb {R}}^m)\big )\) for \(n,m\in {\mathbb {N}}\) and consider rank one operators to be finite elements in \({\mathcal {U}}(E,F)\).

Keywords

Finite elements Orthogonally additive order bounded operators Uryson operators Rank-one operators 

Mathematics Subject Classification

Primary 47H07 Secondary 47H99 

Notes

Acknowledgements

The authors thank the referee for valuable remarks and suggestions, which made the presentation shorter and more precise. M. A. Pliev was supported by the Russian Foundation of Fundamental Research, the Grant No. 17-51-12064 and by the Ministry of Education and Science of the Russian Federation (the Agreement No. 02.A03.21.0008).

References

  1. 1.
    Aliprantis, C.D., Burkinshaw, O.: Locally Solid Riesz Spaces with Applications to Economics (Mathematical Surveys and Monographs), 2nd edn, vol. 105. American Mathematical Society (2003)Google Scholar
  2. 2.
    Aliprantis, C.D., Burkinshaw, O.: Positive Operators. Springer, Dordrecht (2006)CrossRefMATHGoogle Scholar
  3. 3.
    Ben Amor, M., Pliev, M.A.: Laterally continuos part of an abstract Uryson operator. Int. J. Math. Anal. 7(58), 2853–2860 (2013)CrossRefGoogle Scholar
  4. 4.
    Chen, Z.L., Weber, M.R.: On finite elements in vector lattices and Banach lattices. Math. Nachr. 279(5–6), 495–501 (2006)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Chen, Z.L., Weber, M.R.: On finite elements in sublattices Banach lattices. Math. Nachr. 280(5–6), 485–494 (2007)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Chen, Z.L., Weber, M.R.: On finite elements in lattices regular operators. Positivity 11, 563–574 (2007)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Feldman, W.A.: Lattice preserving maps on lattices of continuous functions. J. Math. Anal. Appl. 404, 310–316 (2013)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Feldman, W.A.: A characterization of non-linear maps satisfying orthogonality properties. Positivity 21(1), 85–97 (2017)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Gumenchuk, A.V., Pliev, M.A., Popov, M.M.: Extensions of orthogonally additive operators. Math. Stud. 41(2), 214–219 (2014)MathSciNetMATHGoogle Scholar
  10. 10.
    Hahn, N., Hahn, S., Weber, M.R.: On some vector lattices of operators and their finite elements. Positivity 12, 485–494 (2008)MATHGoogle Scholar
  11. 11.
    Krasnosel’skij, M.A., Zabrejko, P.P., Pustil’nikov, E.I., Sobolevskij, P.E.: Integral Operators in Spaces of Summable Functions. Noordhoff, Leiden (1976)CrossRefGoogle Scholar
  12. 12.
    Kusraev, A.G.: Dominated Operators. Kluwer Academic, Dordrecht, Boston, London (2000)CrossRefMATHGoogle Scholar
  13. 13.
    Kusraev, A.G., Pliev, M.A.: Orthogonally additive operators on lattice-normed spaces. Vladikavkaz Math. J. 3, 33–43 (1999)MathSciNetMATHGoogle Scholar
  14. 14.
    Kusraev, A.G., Pliev, M.A.: Weak integral representation of the dominated orthogonally additive operators. Vladikavkaz Math. J. 4, 22–39 (1999)MATHGoogle Scholar
  15. 15.
    Luxemburg, W.A.J., Zaanen, A.C.: Riesz Spaces, vol. 1. North Holland, Amsterdam-London (1971)MATHGoogle Scholar
  16. 16.
    Makarov, B.M., Weber, M.: On the representation of vector lattices I (Russian). Math. Nachr. 60, 281–296 (1974)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Malinowski, H., Weber, M.R.: On finite elements in \(f\)-algebras and product algebras. Positivity 17, 819–840 (2013)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Mazón, J.M., de León, S.Segura: Order bounded orthogonally additive operators. Rev. Roum. Math. Pures Appl. 35(4), 329–353 (1990)MathSciNetMATHGoogle Scholar
  19. 19.
    Mazón, J.M., de León, S.Segura: Uryson operators. Rev. Roum. Math. Pures Appl. 35(5), 431–449 (1990)MATHGoogle Scholar
  20. 20.
    Mykhaylyuk, V., Pliev, M., Popov, M., Sobchuk, O.: Dividing measures and narrow operators. Stud. Math. 231, 97–116 (2015)MathSciNetMATHGoogle Scholar
  21. 21.
    Orlov, V., Pliev, M., Rode, D.: Domination problem for \(AM\)-compact abstract Uryson operators. Arch. Math. 107(5), 543–552 (2016)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Pliev, M.: Domination problem for narrow orthogonally additive operators. Positivity 21(1), 23–33 (2017)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Pliev, M., Fang, X.: Narrow orthogonally operators in lattice-normed spaces. Sib. Math. J. 58(1), 134–141 (2017)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Pliev, M., Popov, M.: Narrow orthogonally additive operators. Positivity 18(4), 641–667 (2014)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Pliev, M., Popov, M.: On extension of abstract Urysohn operators. Sib. Math. J. 57(3), 552–557 (2016)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Pliev, M.A., Weber, M.R.: Disjointness and order projections in the vector lattices of abstract Uryson operators. Positivity 20(3), 695–707 (2016)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Vulich, B.Z.: Introduction to the Theory of Partially Ordered Spaces. Wolters-Noordhoff, Groningen (1967)Google Scholar
  28. 28.
    Weber, M.R.: On finite and totally finite elements in vector lattices. Anal. Math. 21, 237–244 (1995)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Weber, M.R.: Finite Elements in Vector Lattices. W. de Gruyter GmbH, Berlin/Boston (2014)CrossRefMATHGoogle Scholar
  30. 30.
    Zaanen, A.G.: Riesz Spaces II. North Holland, Amsterdam (1983)MATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Southern Mathematical InstituteRussian Academy of SciencesVladikavkazRussia
  2. 2.RUDN UniversityMoscowRussia
  3. 3.Department of Mathematics, Institute of AnalysisTechnical University DresdenDresdenGermany

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